Perfect-basis realization for strongly positive parabolic bicrystals

Let $({\bf X},{\bf p},f,\Theta)$ be a strongly positive parabolic $(U,\chi^{st})$-linear bicrystal of type $w_P$. A based $\mathfrak{g}^\vee$-module is a pair $(V_\Theta,{\bf B}_\Theta)$ consisting of a $\mathfrak{g}^\vee$-module and a chosen basis. Then there exist a based $\mathfrak{g}^\vee$-module $(V_\Theta,{\bf B}_\Theta)$ and a $\mathfrak{g}^\vee$-linear map

$$h:V_\Theta\to \mathbb{C}[\widehat {G^\vee/P^\vee}]$$

such that the associated normal crystal $\mathcal{B}(V_\Theta,{\bf B}_\Theta)$ is isomorphic to $\mathcal{B}_{f,\Theta}$, and $h({\bf B}_\Theta)$ is a perfect basis for $\mathbb{C}[\widehat {G^\vee/P^\vee}]$.

Perfect-basis realization conjecture. Such a based module and linear map exist.

The source presents this as an equivalent reformulation of the preceding conjecture and suggests that $V_\Theta$ should be the coordinate algebra of an affine cone over a projective $G^\vee$-variety. It also notes that the Schubert-cell case supplies a suitable candidate, while not every $G^\vee$-variety is expected to arise this way.